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Objectives: 1. To add, subtract, multiply, and divide rational expressions 2. To find the partial fraction decomposition of a rational expression • • • • • • Assignment: P. A43: 35-42 S P. A44: 43-52 S P. 539: 15-28 S P. 539: 41-44 S P. 540: 61-64 S HW Supplement You will be able to add, subtract, multiply, and divide rational expressions Find the least common multiple between 2275 and 2156. Find the least common multiple between x2 – 4x and x3 – 8x2 + 16x. To find the least common multiple of two algebraic expressions: 1. Factor each expression into primes 2. Multiply the prime factors to the highest power that they occur in either expression Simplify: x2 2 x Multiply: 3x 3x 2 x 2 x 20 2 x 4x 5 3x Multiplying rational expressions is just like multiplying any other fractions: top1 top2 top1 top2 bottom1 bottom2 bottom1 bottom2 Of course, you’ll need to factor and simplify as needed. One more thing: a b (b a ) Multiplying rational expressions is just like multiplying any other fractions: Divide: 7x x2 6x 2 2 x 10 x 11x 30 Dividing rational expressions is just like dividing any other fractions. Just multiply the first by the reciprocal of the second: top1 top2 top1 bottom2 bottom1 bottom2 bottom1 top2 Again, you’ll need to factor and simplify as needed. Dividing rational expressions is just like dividing any other fractions. Just multiply the first by the reciprocal of the second: Perform the indicated operation. 2 x 2 10 x x 3 2 2 x 25 2 x 6 x 2 x 15 2 (3 x 5 x) 2 4x Add: 2x 5 x6 x6 Add: 2x 5 x6 Add: 2x 5 x 6 x 3 Adding and subtracting rational expressions is as painless as adding and subtracting any other fractions. First you need to get a common denominator, then you just add the tops. top1 top2 top1 bottom2 top2 bottom1 bottom1 bottom2 bottom1 bottom2 To get the common denominator, you should factor first and choose the LCM. Adding and subtracting rational expressions is as painless as adding and subtracting any other fractions. First you need to get a common denominator, then you just add the tops. top1 top2 top1 bottom2 top2 bottom1 bottom1 bottom2 bottom1 bottom2 Or you could just multiply each denominator and simplify twice as much latter. Perform the indicated operation. x 5 x 2 x 12 12 x 48 x 1 6 x2 4x 4 x2 4 You will be able to find the partial fraction decomposition of a rational expression Simplify: 3 2 3( x 4) 2( x 2) x2 x4 ( x 2)( x 4) ( x 2)( x 4) 3x 12 2 x 4 ( x 2)( x 4) x 8 ( x 2)( x 4) x8 x2 6 x 8 Now suppose that, for whatever reason, we wanted to reverse this process. That would be called Partial Fraction Decomposition. It’s like working an addition or a subtraction problem backwards to figure out the original problem. Write the partial fraction decomposition of: x 8 x8 x 2 6 x 8 ( x 2)( x 4) Factor the denominator x8 A B x2 6 x 8 x 2 x 4 Write each factor as a separate fraction with a generic numerator x 8 A( x 4) B( x 2) Multiply both sides by the LCD. This is called the Basic Equation. When all we have are linear factors, solving the basic equation is a matter of choosing convenient values for x and solving for A or B. Solve the basic equation for A: x 8 A( x 4) B( x 2) 2 8 A(2 4) B(2 2) Let x = −2. Solve for A. 6 2A 3 A We could have chosen any value for x, since the equation is true for all values, but when x = −2, B disappears like mathemagic. Solve the basic equation for B: x 8 A( x 4) B( x 2) 4 8 A(4 4) B(4 2) Let x = −4. Solve for A. 4 2B 2 B Thus, the partial fraction decomposition is: x8 3 2 x2 6 x 8 x 2 x 4 To decompose a fraction into partial fractions: 1. If the fraction is improper (n ≥ d), divide and apply the following steps to the remainder. 2. Factor the denominator, not the numerator. 3. For each linear factor in the denominator, write the partial fractions as: A B a1 x b1 a2 x b2 Z an x bn 4. Now multiply each side of the equation by the LCD to obtain the basic equation. To solve the basic equation: 1. Substitute a convenient value for x to make one or more of the numerator values (A, B, C, …) equal zero. 2. Repeat until you find all numerator values. 3. Write the partial fractions using the new numerator values. Write the partial fraction decomposition of: 3x 2 x 5 x3 2 x 2 x When you have repeated linear factors of the form (ax + b)n: 1. You have to include n partial fractions of the form: A1 A3 An A2 ax b (ax b) 2 (ax b) 3 (ax b) n 2. When solving the basic equation, substitute in all known numerator values. Choose other convenient values of x to find the remaining numerator values. Write the partial fraction decomposition of: x2 x2 4x 3 Write the partial fraction decomposition of: 12 x 3 10 x 2 Write the partial fraction decomposition of: 2 x3 x 2 7 x 7 x2 x 2 Partial fraction decomposition may seem like a pointless exercise, but it’s not. Your future calculus teacher wanted me to pre-teach this since there are some functions that you cannot integrate unless you decompose them first into partial fractions. Also, we only tackled the linear factors; look forward to the quadratic ones in the future. Objectives: 1. To add, subtract, multiply, and divide rational expressions 2. To find the partial fraction decomposition of a rational expression • • • • • • Assignment: P. A43: 35-42 S P. A44: 43-52 S P. 539: 15-28 S P. 539: 41-44 S P. 540: 61-64 S HW Supplement